Gaussian fluctuations of differential observables of stationary lattice fields

Abstract

We study Gaussian fluctuations of differential observables of centered stationary random fields on the discrete $d$-dimensional torus of mesh $1/N$. Our main result is a Central Limit Theorem for the fields $f\mapsto N^{-d/2-m}\sum_x\phi_x^N(Pf)(x/N)$, where $P$ is a scalar homogeneous constant-coefficient differential operator of order $m\geq0$ with Fourier symbol $p$, e.g., $P=\Delta$ and $m=2$. Under suitable assumptions on the spectral scaling limit $\gamma$ of the covariance and the normalized cumulants of order $r\geq3$, we prove convergence in law in a negative Sobolev space to a centered Gaussian random distribution with covariance multiplier $|p(k)|^2\gamma(k)$. The two factors separate the effects of the operator and the field and, notably, differentiation can compensate a low-frequency spectral singularity. For the lattice Gaussian Free Field, this gives a projected white-noise limit for the gradient. The proofs use Fourier analysis and the method of moments and cumulants, together with a tightness argument. Finally, we give two sufficient criteria for the cumulant assumption, namely a tree--graph bound and the Dobrushin uniqueness condition for finite-range Gibbs fields, and recover the white-noise limits of the centered Ising magnetization and Potts colour fluctuation fields at high temperature.